Shaping and finishing of aspherical optical surfaces

Shaping and Finishing of

Aspherical Optical Surfaces

Shaping and Finishing of

Aspherical Optical Surfaces

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus Prof.ir. K.F. Wakker, in het openbaar te verdedigen ten overstaan van een commissie,

door het College voor Promoties aangewezen, op maandag 21 september 1998 te 10:30 uur

door

Oliver Wolfgang F€HNLE

Diplom Physiker, Ruprecht-Karls-UniversitŠt, Heidelberg geboren te Mannheim, Duitsland

Prof.dr.ir. H.J. Frankena

Samenstelling promotiecommissie:

Rector Magnificus, vorzitter

Prof.dr.ir. H.J. Frankena Technische Universiteit Delft, promotor Prof.dr.ir. J.J.M. Braat Technische Universiteit Delft

Prof.dr.ir. M.P. Koster Universiteit Twente

Prof.dr.ir. P. Kruit Technische Universiteit Delft Prof.dr. J.C. Lambropoulos University of Rochester, U.S.A. Prof.dr.ir. P.H.J. Schellekens Technische Universiteit Eindhoven

Dr.-Ing. V. Sinhoff Fraunhofer Institut Produktionstechnologie, Aachen, Duitsland

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Advanced School for Computing and Imaging

This work was carried out in graduate school ASCI. ASCI dissertation series number 36.

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to my wife Roberta to my son Alex

Contents

1.2 Abrasive optical fabrication techniques 6 1.2.1 Material removal processes 6 1.2.2 Shaping methods 16 1.2.3 Existing OFTÕs 18 1.3 Scope of the research 19 1.4 Outline of this thesis 20

2. Non-circular Ring Surfacing (NRS) 2 3 2.1 Mathematical description 23

2.2 Functioning principle 28 2.3 Producible surfaces 31

2.3.1 Shape 31 2.3.2 Size 35

2.4 Generation of conic surfaces 41 2.4.1 Using an elliptical Path 42 2.4.2 Optimization 56

2.4.3 Summary 57 2.5 Accuracy considerations 59

2.5.1 Compensation 60 2.5.2 Error propagation 62

2.5.3 Relative speed along the Curve 79 2.5.4 Summary 83

3. FAUST 8 9 3.1 Characteristics 89 3.1.1 Functioning principle 89 3.1.2 Production steps 90 3.2 Accuracy analysis 93 3.2.1 Compensation techniques 94 3.2.2 Error propagation 98 3.3 Process simulation 99 3.4 Machine design 104 3.5 Conclusions 107 4. WAGNER 109 4.1 Characteristics 109 4.1.1 Functioning principle 109 4.1.2 Production steps 111 4.2 Accuracy analysis 112 4.3 Machine designs 113 4.3.1 Machining head 113

4.3.2 Generation of arbitrary shapes 115 4.3.3 Generation of conic surfaces 118 4.3.4 Process simulation 125

4.4 Example: generation of an off-axis hyperboloidal surface 128 4.5 Conclusions 132

5. Experiments on novel finishing techniques 1 3 3 5.1 Fluid Jet Polishing (FJP) 133

5.1.1 Characteristics 133 5.1.2 Experiments 135 5.1.3 Feed controlled FJP 142 5.1.4 Conclusions 146

5.2 Loose abrasive Line-contact Ductile Grinding (LLDG) 148 5.2.1 Characteristics 148

5.2.2 Experiment 148 5.2.3 Conclusions 149

6. General conclusions and suggestions 1 5 1 6.1 Conclusions 151 6.2 Suggestions 155 6.2.1 NRS 155 6.2.2 FAUST 155 6.2.3 WAGNER 156 6.2.4 FJP 158

Appendix A Conic surfaces 161 A.1 Derivation of the function f_c(r) 161 A.2 Function characteristics 164

Appendix B Intersection curve shape (Path) for on- and off-axis conic surfaces 1 7 1 B.1 Introduction 171

B.2 Paraboloidal surfaces 173 B.3 Ellipsoidal surfaces 174 B.4 Spherical surfaces 175 B.5 Hyperboloidal surfaces 176

Appendix C Applying an elliptical Path with a negative off-axis parameter 1 7 9

C.1 Applying an elliptical Path 179

C.1.1 The projection of the Path onto the t, y_r plane is an oblate ellipse (u >v cosa) 181 C.1.2 The projection of the Path onto the t, y_r plane is a circle (u =v cosa) 182

C.1.3 The projection of the Path onto the t, y_r plane is a prolate ellipse (u <v cosa) 182

C.2Radial distance of a point located on an elliptical Path 187 C.3 The generation of off-axis parts of semi-ellipsoids 189

Appendix D Determination of the best starting surface 1 9 1 D.1 Introduction 191 D.2 Procedure 192 D.3 Conic surfaces 196 D.4 Example 198 D.5 Conclusions 198 Symbol list 2 0 1 References 2 0 9 Summary 2 2 7 Samenvatting 2 2 9 Zusammenfassung 233 Riassunto 2 3 7 Acknowledgements 2 4 1 Biography 2 4 2

Chapter 1

Introduction

As an introduction to this thesis, a methodological analysis is given of Optical Fabrication Techniques (OFT) that apply an abrasive contact between tool and workpiece for the generation of refractive and reflective (a)spherical optical components, indicating their functioning elements, and discussing existing OFTÕs. The goal of OFTÕs is the production of optical components (e.g. lenses). Section 1.1 discusses characteristic features of an optical component: its optical surfaces [Subsec.1.1.1], the material it is produced from [Subsec.1.1.2], and its surface shapes [Subsec.1.1.3]. Subsequently, Section 1.2 gives a brief introduction in the currently existing OFTÕs analysing their two functioning elements: the used shaping methods [Subsec.1.2.1], and the applied material removal processes [Subsec.1.2.2]. Subsection 1.2.3 then summarizes the available OFTÕs. Finally, the scope of the presented research is explained [Sec.1.3] and the structure of this thesis is outlined [Sec.1.4].

1.1 Optical components

This thesis deals with different abrasive fabrication techniques for the generation of refractive and reflective (a)spherical optical components. In this section, characteristic features of refractive and reflective optical components will be discussed [Subsec.1.1.1] followed by a description of the used materials [Subsec.1.1.2] and a discussion of the use of aspherical optical surfaces [Subsec.1.1.3].

1.1.1 Characteristics

Optical components (e.g. lenses or mirrors) are used in optical systems for the manipulation of the propagation of light rays. They influence the following properties of an incident light beam: the phase and thereby the direction of propagation, the intensity and the polarization state. This influencing depends on the characteristics of the optical components as listed in TableÊ1.1. All these characteristics have to be defined during the optical design and have to be realised during the production of the optical component. This production process can be divided into two parts: the chemical and physical production of the material and the mechanical shaping of the component. In the optical design and the production of the material most of the characteristics of the component are dealt with in such a way that optical fabrication techniques only have to control the shape accuracy, the surface roughness and the amount of introduced subsurface damage (SSD).

Optical components require a high shape accuracy (to ensure a high imaging performance), smooth surfaces (to avoid scattering), and a minimum of subsurface damage (a.o. to avoid deterioration if used with high power laser beams). While usually low frequency shape errors are addressed in terms of a peak-to-valley shape accuracy and high frequency shape errors are expressed in an rms roughness, the measurement of the power density spectrum (PDS) of an optical surface shape gives information about the whole frequency spectrum of this shape, including its mid frequency errors [Wol96, Aik95, Law95, Har95]. The amount of subsurface damage (SSD) is measured in terms of thickness of the distorted layer below the surface [Sch95, Kran94] and the amount of surface stress [Lam96a, Lam96b, Rup87, Nik85, Twy88].

The optical performance of optical components therefore depends very much on the possibilities of the fabrication technique with which they have been manufactured.

Table 1.1 Dependence of the optical performance of an optical component on its characteristic features (based on [San87])

Characteristics of optical components

Influence to its optical performance

angle of incidence misalignment of an optical component in an optical system causes optical aberrations (e.g. coma)

diameter determines diffraction limited size of the focus

(determined by the radius of the Airy disc [Sal91]) surface shape spherical surfaces cause geometrical aberrations shape accuracy inaccurate shape causes geometrical aberrations

surface roughness causes scatter and is responsible for the loss of imaging contrast

SSD that was generated during the fabrication (micro-cracks, stress, physical and chemical changes of the material, etc.)

causes light loss; limits lifetime and increases absorption by the optical component

absorption can cause unwanted light loss in the optical system reflection can cause unwanted light loss in the optical system and

produce ghost images

transmission changes the intensity of the light ray dispersion causes chromatic aberrations (dispersion)

refractive index inhomogeneities of the refrective index cause distortion to the phase of the light ray

sensitivity to environmental influences (temperature, mechanical vibrations, pressure, radiation, humidity, contamination (e.g. dust))

limits lifetime of the optical component; changes amount of SSD, roughness and shape accuracy and causes therefore the effects listed there

mechanical stress the amount of stress determines polarization effects strength depends inter alia on the roughness and has no direct

influences; it is responsible for the deformability and crack initiation of the optical surface

density of the material inhomogeneities cause unwanted changes to absorption and refractive index and change the phase and intensity of the light ray

chemical and physical inhomogeneity of the material

inhomogeneities cause changes in refractive index, reflection, the phase of the light ray, etc.

1.1.2 Optical materials

Optical fabrication techniques have to deal with the physical and chemical surface characteristics of the used materials. These characteristics are determined by their structure and chemical composition as well as by the environmental conditions in which the optical component is used (e.g. vacuum or gaseous environment). A discussion of the characteristics of optical materials was given by Karow [Kar93(a)]. Optical materials can be divided in metals (e.g. aluminum, beryllium), optical plastics (e.g. TPX, SAN), optical crystals (e.g. alkaline earth fluorides, laser crystals), IR materials (e.g. metallic germanium, silicon, gallium arsenide) and glasses [Kar93(a)]. The research reported in this thesis focusses on the generation of optical surfaces in brittle materials (e.g. glass). Therefore, in the following the major characteristics of glass are summarized.

Glass The composition and properties of optical glasses are investigated in the field of glass science [Bac95a, Izu86, Sch88a, Dor94a, Vog92a]. Glass is an amorphous and vitreous solid and has a non-thermal equilibrium chemical structure. It is produced by increasing the viscosity of a liquid faster than that a crystalline structure can be formed. This can be done e.g. by cooling [Pfa89a], or by chemical reaction (sol-gel forming [Vog92b, Dis83]). Consequently, there exists no long-range structural order [Goo86, Sch88b].

Various glass models have been developed [e.g. Zac32, Die48, Sme49, Wey59, Goo86, Stev60, Leb21, Hug55 and Uhl83] explaining the characteristics and the behaviour of different kinds of glass (e.g. silicat glasses, chalcogenide glasses and metallic glasses) and have been reviewed in detail by Scholze, Vogel, Izumitani, Bach and Doremus [Bac95a, Izu86, Sch88a, Dor94a, Vog92a].

Glasses can have very high dopant levels which are used to control their optical behaviour (e.g. the refractive index), can accurately be given a specific shape (no crystal structure) and can be transparent in the infrared, visible and ultraviolet regions of the spectrum.

Optical fabrication has to deal with glass surfaces and its sub-surface regions (several mm deep) [Dor94b, Sch88c, Ern72, Lie90]. Glass surfaces consist of various chemical substances and a hydrated layer (e.g. SiOH groups for silicat glasses). The major characteristics that have to be considered are their mechanical properties [Ern77] (e.g. fracture toughness [Law90, Hol86, Swa79, Wie69]), chemical reactibility (e.g. reactions with gases or water [Coo90, Sch82, Mac87, Mul79]) and structural properties (e.g. ion difusion into the glass network [BŠr91, Nog83, Alek94, Gar85, Rav85]).

1.1.3 Aspheres

Apart from the influences of material characteristics, optical components limit the quality of imaging of an optical system because of geometrical aberrations (spherical aberration, coma, etc. [Mal92a]) and diffraction effects [Sal91]. Geometrical aberrations can be avoided by using optical components that have an aspherical shape. Rotationally symmetric aspherical surfaces can be described by the function f(r) (taking the f axis as the axis of revolution), where in principle f(r) is described by a convergent power series of r. Splitting off the expression describing conic surfaces [App.A], one can write it as

f r r p p r k r ( ) = ± + -

(

)

( )

where y(r) again is a power series of r, p is the radius of curvature at the vertex, k is the conic constant [App.A, Nau92, Mal92b], and the plus- and minus signs refer to concave and convex surfaces, respectively.

The use of aspherical optical components in an optical system minimizes the number of lenses required to obtain a certain image quality [Sch87]. Therefore, such an optical system needs less space, has less weight, and, as there are less optical surfaces involved, causes less absorption.

Important fields of application for aspherical optical components are

¥ astronomy: telescopes, space research ¥ medical industry

¥ laser industry

¥ photo industry: cameras ¥ lithographic technology

¥ CD player and CD-ROM technology ¥ armaments industry.

In order to fulfill the high demands on refractive and reflective optical components, it is necessary to develop accurate fabrication techniques that generate aspherical optical surfaces in brittle materials (e.g. glass) with a high shape accuracy (» 30 nm PV), a low level roughness (»Ê0.5 nm rms) and a minimum amount of subsurface damage (depth < 50 nm).

1.2 Abrasive optical fabrication techniques

Overviews of available OFTÕs have been given by Heynacher, Marioge, Franse and FŠhnle [Hey78, Mar84a, Mar84b, Fra90 and FŠh98f].

This section presents a methodological analysis of OFTÕs that apply an abrasive contact between tool and workpiece for the generation of rotationally symmetric (a)spherical surfaces, consequently omitting the generation of aspheres by the use of moulding [Par81, Zwi85, Zwi86, Bra85] or by the use of rotating fluids [Gob87, Gib95, Borr95, Nin96, Thi98].

It focusses on OFTÕs for the generation of optical components in brittle materials (e.g. glass). The fabrication of aspherical optical surfaces in brittle materials consists in general of at least two subsequent steps. The first one is a rough shaping mode (e.g. loose or bound abrasive brittle mode grinding [Bac95b, Kir84, Phi77, Gol91, Zha94, Hue76, Rup72]), where the aspherical shape of the surface is generated. Here, the material removal is caused by brittle fracture. This production step is characterized by a high material removal rate and leaves a rough surface with an highly stressed layer of subsurface damage (e.g. loose abrasive grinding: 1-10 mm depth [Edw87, Par90]). The second production step is the finishing mode. Here the roughness, surface stress and the subsurface damage, that were generated during the rough shaping mode, have to be removed and the shape accuracy has to be improved.

The Optical Fabrication Techniques (OFTÕs) discussed in this thesis belong to the finishing mode and use as a starting point a roughly shaped optical surface in brittle material. These OFTÕs comprise the following two elements: the used Material Removal Process [Subsec.1.2.1] and the applied Shaping Method [Subsec.1.2.2]. Although these two elements are often interrelated, they will be treated separately in the following two subsections. Finally, existing OFTÕs will be summarized [Subsec.1.2.3].

1.2.1 Material removal processes

The surface quality (level of roughness and amount of subsurface damage) of a generated rough starting surface (either a roughly shaped aspherical surface or a best starting surface [App.D]) can be improved by various Material Removal Processes, which will be discussed in this subsection.

In general the following Material Removal Processes are used for the finishing of optical surfaces: fresh feed polishing, ductile grinding, chemo-mechanical polishing, bowl feed

polishing, elastic emission machining, magnetorheological finishing (MRF), flame and laser fire polishing, ion beam polishing, abrasive slurry jet machining, and plasma assisted etching. Except from bowl feed polishing, which is currently only used for the finishing of flat surfaces, all other methods can also be used for the finishing of aspherical surfaces.

In the following, the major characteristics of these Material Removal Processes are summarized and, for a better comparison, are listed in Table 1.2. There, a short description of each method is given, together with the lowest level of roughness currently reached (in glass).

A Fresh feed polishing

The traditional fabrication technique that is used for the generation of flat and spherical optical surfaces is loose abrasive brittle mode grinding [Bac95b, Lia97] followed by loose abrasive fresh feed polishing [Fie95a, Bac95c, Nar88, Lam98].

A loaded mould (usually pitch is used for the tool surface [Lin86]), which has the inverse spherical shape of the one that is generated, is moved over the rotating workpiece. The tool and the workpiece form an area-contact and the material removal is carried out by abrasive particles (e.g. metal-oxydes), which are moved randomly in the slurry (usually water) between tool (usually pitch [Tes90]) and workpiece. The polishing process is based upon a complex interrelationship of a large number of variables [San87b, Fie88], the most important of which are

¥ kind and amount of the used suspension and its physical and chemical properties, ¥ kind of used abrasives and their physical (particle shape, hardness) and chemical

properties [Proc86a, Tes90, Sec76, Cou94, Cum95, Par94], ¥ concentration, temperature and chemical reactibility of the slurry,

¥ physical and chemical characteristics of the used tool material [Lei76, Lei93, Lei92], ¥ physical and chemical characteristics of the starting surface [Cum94, Prim81], ¥ physical and chemical properties of the used glass [Eva94],

¥ applied load and relative speed between tool and workpiece, ¥ environmental conditions (e.g. room temperature) [Baa68].

The material removal is carried out through mechanical and chemical effects [Kir94, Coo90, Bro87, Lam96a] and depends on the load and the relative motion between the tool and the workpiece. The local material removal rate dh /dt is usually estimated by PrestonÕs equation [Pre27] as

dh dt c L Q ds dt = Pres , (1.2)

where L denotes the applied load, Q the contact area between tool and workpiece, ds/dt the tool speed parallel to the surface and c_Pres a constant that depends on the process parameters [Bac95d, Fie95b, Lam97]. Conventional polishing is an area-averaging process where each point of the mould gets into contact with each point of the workpiece surface and higher material removal rates occur at local elevations of the workpiece surface. This, together with pitch flow effects, causes a smoothening of the surface. Therefore, the fabrication technique of conventional grinding and polishing is a self-correcting process. Two initially imperfect surfaces, tool surface (pitch) and workpiece surface (glass) change during the fabrication into high quality surfaces (low level roughness) with a high shape accuracy [Jon95]. As this self-correcting process is controlled by the load of the tool and the relative speed between tool and workpiece, it is not necessary to ensure high precision motions of the tool and the workpiece. Fabrication techniques of this kind are referred to as soft-loop fabrication techniques.

For the generation of aspherical surfaces, which require only small deviations from a spherical surface, it is possible to use the self-correcting process of conventional area-contact polishing by controlling pitch-flow effects and pressure distribution over the workpiece surface.

For steeper aspherical surfaces the deviation from a best fitting spherical surface is too large and therefore it is necessary to use fabrication techniques which employ a line or a point contact. Whilst it is still possible to achieve a self-correcting process with fabrication techniques that employ a line contact, it is impossible to achieve it with fabrication techniques that use a point contact. Therefore, fabrication techniques that use a point contact require the same accuracy for the guidance of the tool as it is required for the surface to be generated. The roughness and shape accuracies of the generated surface depend directly on the rigidity of the machine. These fabrication techniques are referred to as hard-loop fabrication techniques (e.g. the single point diamond turning of optical components).

B Ductile grinding

The material removal process of grinding of optical surfaces can be understood by analysing the crack generation beneath a sharp indenter [Lam97, Ham94, Cha93, Aro79, Wie74, Law75, Law80, Bui93, Wil90, Coo90, Bre87, Wal89]. As shown in Fig.1.1, a radial crack is generated, if the load of the sharp indenter (e.g. an abrasive particle) exceeds a critical value. Reducing the load [Fig.1.1 b, c and d], results into a lateral crack. If this lateral crack grows until it reaches the surface, a chip of material is removed (brittle mode material removal). The remaining radial crack determines the depth of the subsurface damage that is introduced to the optical surface.

plastically deformed

zone

radial crack lateral crack

removed chip

(a) (b) (c) (d)

Fig.1.1 Sketch of the material removal in brittle mode grinding or lapping of optical surfaces: (a) increasing the load results in a plastically deformed layer below the point contact, (b) after the maximum load is applied and the radial crack is generated, the load is reduced resulting into a lateral crack (c) that grows with the decreasing load; (d) if the lateral crack reaches the surface, material is removed leaving a pit with a radial crack beneath the surface.

If the introduced energy remains below the energy needed for crack initiation, material can be removed by viscous flow (ductile material removal). Since the energy E_p needed for plastical material deformation is proportional to the deformed volume and the energy E_c required for the

generation of a crack is a function of the area created by crack propagation, the ratio of material removal energies can be written as

E

E d

p

c i

µ , (1.3)

where d_i denotes the penetration depth of the deformation. Therefore, with decreasing depth of intendation plastic flow becomes more favorite as material removal mechanism [Bif91]. Consequently, there is a critical depth of cut in order to achieve a ductile grinding mode. A ductile ground glass surface leaves a smooth surface (better than 7 nm rms roughness) with a subsurface damage depth that is comparable with polishing [Sin92, Sin95, Ben90, Dai95, Sch91, Bla90, Sho95, Wil90, Faw91, Law76, Lam96c, Schi91]. The subsequent polishing step then has to remove the stressed surface layer that origins from the plastical material removal [Gol91, Lam96b] and to reduce the surface roughness.

A ductile removal mode has been achieved within hard-loop fabrication techniques, accurately controlling the relative position and the infeed rate of a tool with respect to the workpiece. This can be done by fixed abrasive precision grinding [Faw91, Bla88, Faw91, Lam96d, Fes98] or by single point diamond turning (SPDT) [Fang97, Klo95, Jea86, Fuc86, Mor88, Myl90, Bas95b, Bla90, Nak90, Bif91, Pau96, Fal87, Shi95, Sai90, Sca90].

Besides, it is possible to achieve a ductile grinding mode with the soft-loop fabrication technique of loose abrasive load controlled grinding [Gol91, Bro88] through control of the physical characteristics of the applied slurry.

C Chemo-mechanical polishing

Glasses are dissolvable e.g. in acid solutions [Lia87] or water. Consequently, such chemical reactions can be used to smoothen optical surfaces. This etching without mechanical contact results in a surface that contains unwanted pits [Lia87, Vog92a]. Therefore, chemical reactions are mainly used to additionally smoothen the surface during lapping or bowl feed polishing of e.g. wafers (which will be described in the following subsection) [Mci80, Nam79, Lin88, OÕHa94, Nan95, Pri91, Zha89, Lia87, Esc87].

D Bowl feed polishing

Bowl feed polishing (also float polishing or superpolishing) is a polishing process usually used to finish ultra smooth flat silicon surfaces for wafers (about 0.12 nm rms [Win92]). This is a loose abrasive polishing process where tool and workpiece are immersed in the polishing slurry. The tool is floating on the slurry and the thickness of the liquid layer between tool (usually pitch or teflon) and the workpiece surface is a multiple of the diameter of the used abrasive particles (usually cerium oxide). After the polishing process is started no slurry is added. Therefore, due to the rotary speed of the workpiece the polishing particles are transported over the edge of the surface and sink down to the bottom of the bowl. In the end, the slurry between tool and workpiece contains no more abrasive material apart from that embedded in the tool surface [Soa94, Win92, Ben87, Bas94, Schm95, Die66, Gun94, Gor81, Wei90, Zak92, Soa90, Sek94].

E Elastic Emission Machining (EEM)

Minimizing the contact area between tool and surface, Elastic Emission Machining (EEM) [Mor87, Mor88, Wan95] makes use of a float-polishing process where the tool (a polyurethane sphere) is floating on the liquid layer containing the abrasive particles. The thickness of this liquid layer amounts to a multiple of the diameter of the abrasive particles and the process-determining parameters include the hydraulic pressure generated by the tool and the kinetic energy of the abrasives. That way, a local float polishing can occur. Surface roughnesses of better than 50 nm PV in single crystal silica have been reported [Mor88].

F MagnetoRheological Finishing (MRF)

Whilst in EEM the tool is not pressing the abrasives directly onto the surface, the tool is abandoned entirely in MagnetoRheological Finishing (MRF) [Lam96e, Kor94a, Kor94b, Gol96, Gol98, Jac96, Jac97, Jac95a, Jac95b, Kor96]. A fluid containing magnetic sensitive particles mixed with polishing compound (cerium oxyde) is locally stiffened by the use of a magnetic field and is thus used for local polishing and shaping. In the zone where the magnetic field is applied, the magnetic sensitive particles force the polishing abrasives against the surface to be polished. MRF is a sub-aperture polishing method that results in polished surfaces with a roughness of about 1 nm rms in glass. Thanks to its kinetic character, no tool wear occurs and the tool cools and removes debris in the process.

G Laser fire polishing

Using a laser beam (e.g. from a CO₂ laser) or a flame, optical surfaces in glass can be smoothened by heating them to approximately their transformation points [Lag94, Bow94, Coe93, Xia83, Tem82, Mas88, Tem82, Xia83]. Laser fire polishing is a trade off between two effects: on one hand a fast heating is necessary to maintain the required surface shape (the surface melts slightly) and on the other hand the surface has to be heated sufficiently to anneal any subsurface damage from the previous rough shaping mode. Therefore, laser fire polishing has to introduce a large amount of energy in a short time into the material, resulting in thermal stress at the optical surface. A surface roughness of 1 nm rms in BK7 has been reported [Lag94].

H Ion beam polishing

In ion beam polishing, the optical surface is exposed to an ion beam whereupon the material is removed by sputtering. The sputtering rate depends on the angle between the incident beam and the local normal to the surface. There are three different applications of ion beam polishing. To smoothen e.g. a diamond film, an optical coating is previously deposited which has the same sputtering characteristics as the surface to be polished. Subsequently, the surface covered with the planarizing film is exposed to the ion beam and material is uniformly sputtered away until the planaryzing film is removed resulting in a uniform smooth surface [Bog92, Gro92, Bov92, Joh83].

Furthermore, starting from a rough flat surface and applying an ion beam exposure using an ion beam that is parallel to the normal to the surface, smoothening of the surface due to the angular dependence of the sputtering rate has been proposed [Car93, Hou76]. Conversely, various publications present experimental data indicating that ion beam machining requires a smooth starting surface to maintain the surface roughness during shaping [e.g. Zar93, Faw94].

Finally, ion beams are used to shape a finished optical surface without increasing its surface roughness. This can be performed locally [Dru95a, Dru93, Hor72, Schr71, Schi94, Kar91, Car95, Dru95b, Faw94] or using a mask for exposure of the optical surface [Mul95]. In special cases, neutral particles can be used instead of ions.

I Abrasive slurry jet machining (ASJ)

Guiding a slurry of high viscosity, containing polymer abrasives, by the use of a nozzle at high pressures (> 50 bar) onto a glass surface, its roughness can be reduced [PCT/US97/03316]. As the polymer molecules (sizes of 10 to 200 mm) hit the surface, they remove the surface roughness by their lateral relative movement with respect to the surface.

J Plasma assisted etching

Optical surfaces can be shaped and polished by Plasma Assisted Chemical Etching (PACE). A small, confined plasma tool is moved over the surface and is locally polishing the material. PACE is a non-contact method that applies plasma assisted etching. Surfaces of 1 nm rms have been reached in silicat glasses [Bol90, Zar93].

Table 1.2

Material removal processes

Finishing- Method

Description

Advantages

Problems

Achieved surface roughness

Fresh feed polishing chemical-mechanical process that reduces roughness, stress and subsurface-cracks from a previously ground surface removes stress; causes minimum subsurface damages; using pitch-flow effects an in-process correction of the generated shape is possible small amount of wear; more than 30 variables influence the process;

0.5 nm rms in BK7:

Ductile grinding (dg)

removal of material is caused by plastic flow and not by brittle fracture; dg is a mechanical process that depends on the elastic constant of the material and on the machining depth; it is possible to achieve a dg mode with single point diamond turning (SPDT), precision grinding, and loose abrasive grinding dg achieves higher material removal rates (

≈

30

µ

m/h)

than fresh feed polishing while it leaves nearly no damaged subsurface layer (same depth as with fresh feed polishing (< 20nm) dg leaves a smooth surface that is highly stressed SPDT and precision grinding: to achieve machining depths of nm-scale a high infeed resolution is necessary: this requires a stiff construction of machinery and/or small infeed speeds; loose abrasive grinding: chemically safe production environment is required 2-3 nm rms in BK7

Chemical polishing alkaline-solution causes material removal; acid-solution replaces the network-formers A) material removal through dissolu- tion of the surface in a chemical fluid( e.g. HF, HNO

3,

or H

2

O)

B) combined with fresh feed or bowl feed polishing A) no mechanical contact; causes no subsurface damage and no plastical deformed surface layer; B) generates a low level roughness and a high shape accuracy combined with a low amount of subsurface damages A)because there is no mechanical contact, the shape accuracy is difficult to improve; products from chemical reactions stick onto the surface and will have to be removed later on; extremely sensi- tive to inhomogeneities of the material B)chemical slurry can break down pad-surface; mechanical contact causes plastically deformed layer A) no data B) chemical bowl-feed polishing of a flat surface in Si results in less than 0.5 nm rms

Bowl feed polishing (float polishing) sample and lap are located in a bowl and submerged in a slurry containing the polish. powder. Sample and lap rotate with the same speed in the same direction. The polishing particles sink away, less and less particles flow between sample and lap until finally the sample floats on nothing but water generates supersmooth, damage free surfaces; in the beginning there is a mechanical contact, but at the end sample and lap are separated only by a layer of water to achieve a high shape accuracy a temperature control is necessary (pitch-flow); complicated to achieve a bowl feed polishing over the whole sur- face

E

E

M

a polyurethane sphere is floating on the liquid layer containing the abrasive particles, the thickness of which exceeds the applied abrasive size; chemical activated local removal of atomic scale since the sphere removes locally material this is a promising method for local float polishing of curved optical surfaces until now, no superpolished (< 0.5 nm rms) have been reached; elastic behaviour and wear rate of the applied polyurethane sphere still need investigation 50 nm PV roughness on flat single crystal silica

M

R

F

within the slurry, magnetic sensitive particles press polishing abrasives against the optical surface.

Thanks to its kinetic character, no tool wear occurs and the tool is cooling and removing debris in process, variable shaping method thanks to its dependence on the applied magnetic field slurry has to contain magnetic sensitive particles, reaction time to the magnetic field and slurry characteristics have to be studied

about 1 nm rms in glass

Laser fire polishing decrease of surface roughness and healing of subsurface damage through heating the surface layer nearly up to the melting point (

≈

1000 ˚C); there are

two possibilities: A) heating the whole surface with a laser beam, or B) heating of a small area of the surface with a focused laser beam no mechanical contact; finishing of various shapes possible; punctual melting using a focused laser beam maintains the shape of the surface high temperature causes: -chemical changes in the material, -chemical reactions between glass and environment, -diffusion of gases, -separation of the glass, -increased corrosion; cooling down process necessary to avoid cracks; enough energy needed to melt the material deep enough to heal sub- surface damage from previous grinding (

≈

10-20

µ

m); generation of stress in the material through

both: heating and afterwards cooling; to prevent cracking a preheating (>700˚) and a cooling down process are necessary flat surface in BK7: 1nm rms

Ion beam polishing sputtering erosion of the surface is a stochastic atom-by-atom removal process; the sputtering yield is a function of the macroscopic angle between the direction of the incident ion flux and the local normal at each surface point no mechanical contact; finishing of various shapes possible [Carter 93]; exact aspherical shaping possible (if a focused ion-beam is used) vacuum needed; sputtering itself is an erosive process; implantation of ions into the material; chemical reactions between ions and atoms of the surface; improvement in roughness after long polishing times if the precise initial surface profile is known; small amount of wear; very sensitive to existing subsurface damages

figuring 5

µ

m

away from a flat ceramic surface 2nm rms roughness was maintained

A

S

J

a slurry of high viscosity containing polymer abrasives is guided by the use of a nozzle at high pressures (> 50 bar) onto a glass surface

no

mechanical

contact

high stream pressures and a slurry of high viscosity are required

no data

PACE

small, confined plasma tool is locally polishing and shaping the optical surface

locally

polishing

method

tool dimension and complicated tool construction limit the finishing of strongly curved optical surfaces flat silicat glass: 1 nm rms

1.2.2 Shaping methods

Shaping Methods in OFTÕs can be classified into two categories: based on tool shape copying methods [Subsec.1.2.2A] and based on surface evolution calculations [Subsec.1.2.2B].

The shaping of aspherical optical surfaces have been reviewed in detail in literature [Mar84b, Koc91, Vin62a, Hey78].

A Shape copying methods

These kinds of abrasive fabrication techniques use shape copying methods for the generation of the required aspherical shape of the workpiece. This can be performed in three different ways: applying an area contact, a line contact or a point contact between tool and surface [Mar84b, Koc91, Vin62a, Hey78].

Shape copying methods using an area contact Applying an area contact, the shape of a tool surface is copied directly into the optical surface. This Shaping Method is usually used with loose abrasive grinding and polishing for the generation of spherical surfaces (as discussed in Subsec.1.2.1A) and can only be used to generate aspherical shapes that have small deviations from a spherical one. To that aim, pitch flow effects and control of process parameters (e.g. length of stroke or load distribution along the tool surface) are employed and the achievable shape accuracy depends strongly on the experience of the technician.

Besides, it is possible to deform a sample (by e.g. vacuum) in a predetermined way. Subsequently, this stressed surface is ground and polished into a flat or spherical shape. Finally, the surface is removed from its deforming mould and the predetermined aspherical surface shape is obtained. In that way also non-rotationally symmetric aspherical surfaces can be generated [Nel80a, Nel80b].

Shape copying methods using a line contact For the generation of steeper aspheres, no shape copying area contact methods can be applied. Consequently, line contact methods have been developed. The aspherical shape is generated by copying the shape of a two-dimensional blade that has the inverse shape of a cross-section of the surface to be generated that contains its axis of symmetry (its meridional line) [Des1638, Hey78]. The workpiece is rotating and the blade is fixed above the workpiece. This, together with tool wear effects results in severe surface shape errors. In addition, shape errors made in the generation of the two-dimensional blade shape are completely passed on to the surface.

To avoid these shape errors for the generation of hyperboloidal surfaces, Wren [Wre1669, Hey78, Twy88a] proposed a method where two rotating cylinders are arranged above each other such that their symmetry axes form an angle and are positioned perpendicular to the symmetry axis of the rotating workpiece (both cylinders axes cross the symmetry axis of the workpiece at different heights). During grinding, both cylinders wear and thanks to that hyperboloidal surfaces are generated.

In order to achieve an advantageous error propagation factor between blade shape and generated surface shape, evolute methods that apply in-process shape corrections of the copied meridional line have been developed a.o. by Mackensen [Hey78, Twy88a].

For the generation of conic surfaces (rotationally symmetric surfaces which have a conic section as generator [App.A]), Vinokur described a line contact method that is based on a curve generator. A rotating tube is fixed above a workpiece in such a way that its symmetry axis crosses the symmetry axis of the rotating workpiece. In that way, the circular cross-section of the tube generates a spherical surface, the radius of which depends on the angle between the two symmetry axes [Subsec.2.5.1, Bech94, Kar93b]. A misalignment of the tube along an axis perpendicular to both symmetry axes results in an off-axis surface the shape of which approximates a conic surface [Vin62a, Che77].

Shape copying methods using a point contact Furthermore, it is possible to copy the predetermined path of a small tool directly into the material (e.g. single point diamond turning SPDT [Fang97, Klo95, Jea86, Fuc86, Mor88, Myl90, Bas95b, Bla90, Nak90, Bif91, Pau96, Fal87, Shi95, Sai90, Sca90] or precision grinding [Faw91, Bla88, Faw91, Lam96d]). Tool and workpiece are in point contact and the tool path along the meridional line results directly in the generated surface shape. These fabrication techniques impose high demands on the accuracy of the guidance of the tool and require very rigid machine constructions which have to be effectively isolated from external vibrations. Therefore, the application of this kind of Shaping Methods was only possible once the required hard-loop machining tool accuracies where achieved enabling a ductile grinding of brittle materials .

B Shaping methods based on surface evolution calculations

Shaping Methods in which the aspherical surface shape is determined by wear calculations can be divided into two categories: Shaping Methods applying an area contact whose shape is calculated in advance, and Shaping Methods where the material removal rate is calculated along the path of a small tool that applies a small area contact.

Varying in a predetermined way the shape and the amount of the contact area between spherical or flat mould surface (e.g. made from pitch) and the workpiece surface that is loose abrasive fresh feed polished, aspherical surfaces can be generated (e.g. segmented pad polishing [Twy88a, Cha91, Jon82]). Ion beam figuring, using exposure masks, is based on the same functioning principle [Subsec.1.2.1H].

Within stressed lap shaping [And91, Mar90, Rup93] a relatively large and stiff polishing tool is used for area contact loose abrasive fresh feed polishing of large (e.g. 8 m in diameter [And91]) mirrors. The shape of the tool is changed actively in-process in order to generate the required aspherical surface shape.

In computer controlled polishing (CCP [Bro96, Gig89, Gig91, Fun93, Tho86, Por93, Wag74, Baj76, Jon90, Dou87, Jon91, Jon95, Asp72]) a small polishing tool is moved under computer control over the surface to be generated. The surface is loose abrasive fresh feed polished. CCP is characterized by the iteration of in-process shape measurements and tool-path corrections. Although fresh feed polishing is usually employed, there exist other Material Removal Processes that also can be applied locally using wear calculations (e.g., MRF [Subsec.1.2.1F]).

1.2.3 Existing OFTÕs

In this subsection, modern OFTÕs are summarized in relation to their Material Removal Processes and Shaping Methods.

The state-of-the-art of the fabrication techniques for the generation of optical surfaces of revolution is feed-controlled precision machining (single-point diamond turning or fixed abrasive grinding [Subsec.1.2.1B]), in which tool and workpiece are in point contact, while the shape is generated by copying the tool-path [Subsec.1.2.2A]; the material removal process is brittle mode grinding followed by ductile mode grinding. Subsequently this surface is being finished by CCOS (computer controlled optical surfacing) in which a small polishing tool [Subsec.1.2.1A]) is moved over the surface to be generated by computer control. CCOS is characterized by the iteration of in-process shape measurements and tool-path corrections [Subsec.1.2.2B].

Alternatively, the optical surface can be finished applying a local MRF based on wear calculations [Subsec.1.2.1F, Subsec.1.2.2B], or previously loose abrasive fresh feed polished spherical surfaces can be locally shaped into an aspherical shape applying ion beam shaping [Subsec.1.2.1H, Subsec.1.2.2B].

1.3 Scope of the research

The purpose of this research was to develop a new Optical Fabrication Technique (OFT) for the generation of rotationally symmetric aspherical surfaces in brittle materials. In general, the research in the field of optical fabrication can be divided into two categories:

· investigations concerning the developement of new OFTÕs (developing new Shaping Methods or Material Removal Processes, or creating new combinations of existing Shaping Methods and Material Removal Processes)

and

· investigations concerning the optimization of machine designs that are based upon existing OFTÕs (e.g. the development of tool error compensation techniques or the improvement of machine tool accuracy).

The goal of the investigation reported in this thesis was the development of a new OFT rather than the optimization of a machine tool that applies an existing OFT (e.g. precision grinding or diamond turning of aspherical optical components).

During the analysis of existing OFTÕs we focused on non-traditional finishing research in order to develop a new OFT that enables the use of self-correcting traditional OFTÕs for the generation of highly accurate aspherical surfaces. Combining the characteristics of conventional grinding and polishing with the possibilities offered by modern machining tools, a new OFT, named FAUST [Ch.3], has been developed.

During theoretical analysis and simulations of the FAUST functioning principle, the idea of a second OFT, indicated as WAGNER [Ch.4], was generated and led to an extension of the project.

Investigating the finishing of conformal optics, a novel sub-aperture polishing method, FJP [Ch.5], was invented and its experimental proof together with the investigation of its Material Removal Process was included into the research project.

1.4 Outline of this thesis

This thesis comprises six chapters and four appendices. The results of an investigation in the field of non-traditional finishing research of aspherical optical surfaces in brittle materials are presented.

Chapter 1 gives a methodological review of optical fabrication techniques (OFTÕs) for aspherical optical surfaces indicating their functioning elements and discussing existing OFTÕs.

Chapter 2 presents a new Shaping Method, NRS (Non-circular Ring Surfacing) for the generation of aspherical surfaces of revolution. The producible shapes are discussed, an accuracy analysis is presented and the generation of conic surfaces is investigated showing that, although NRS is essentially a shape copying method, different surface shapes can be generated using the same ÒmasterÓ.

In Chapter 3, a new OFT, named FAUST (Fabrication of Aspherical Ultraprecise Surfaces using a Tube), is presented. FAUST is based upon NRS and is a line contact method for the generation of aspherical surfaces and resembles a modified curve generator. Its functioning principle together with possible Material Removal Processes are analysed, a process simulation is given and a first FAUST machine design is discussed.

Chapter 4 presents another new OFT, with the acronym WAGNER (Wear-based Aspherics Generator using a Novel Elliptical Rotator), that again is based upon NRS. WAGNER is a load controlled point contact method for the generation of aspherical optical surfaces. Its functioning principle is analysed and an accuracy analysis together with a process simulation are presented. The machine design of a WAGNER machine tool for the generation of all conic surfaces is discussed and first experimental results are shown.

In Chapter 5, the machining of optical surfaces is investigated. The experimental results of a load controlled line contact ductile grinding of BK7 are reported and a new sub-aperture finishing method, FJP (Fluid Jet Polishing) is presented. Initial experimental results are given and the Material Removal Process of FJP is discussed.

Subsequently, Chapter 6 presents conclusions and suggestions for future research discussing possible applications of the mentioned novel optical fabrication techniques.

In Appendix A, the mathematical description of conic surfaces is derived. It is shown that the minimum distance of the generating point of a paraboloidal, ellipsoidal or hyperboloidal surface to the axis of symmetry of the surface yields the radius of curvature at the vertex.

Appendix B proves that for the generation of all conic surfaces, elliptical intersection curves are required.

Subsequently, in Appendix C, the influences of a negative off-axis parameter are investigated. It describes the transition from the generation of convex surfaces to the generation of concave surfaces using the same production parameters and only varying the off-axis distance between tool and rotary axis of the workpiece.

Finally, Appendix D presents a method to determine the best starting surface for the generation of aspherical optical surfaces of revolution.

Chapter 2

Non-circular Ring Surfacing (NRS)

In this chapter, the functioning principle for the generation of aspherical surfaces of revolution using non-circular shaped rings (Non-circular Ring Surfacing, NRS) is presented. We describe its characteristics and discuss the producible shapes together with an accuracy analysis.

2.1 Mathematical description

Non-circular Ring Surfacing (NRS [FŠh96a, FŠh96c, FŠh97b]) is a shaping method that can be combined with different material removal processes. It is essentially a shape copying method that enables the generation of rotationally symmetric surfaces. In this section, its mathematical description is presented.

Consider a rotationally symmetric surface in the t, y_r, f space that is given by a function g(t,y_r). Let f(r) describe the cross-section of this surface in a plane containing its f-axis of symmetry (with r2_=y

r

2_+t2_{). For each radial section (r in the range of D £ r £ r}

m and |D| < |rm|) of a convex or concave (as seen in the negative f direction) surface of revolution g(t, y_r), there is a unique intersection curve obtained by intersecting the surface with a plane parallel to the y_r axis that in the t, y_r, f coordinate system contains the points P_D = {D, 0, f(D)} and P_m = {r_m, 0, f(r_m)}. Now, two Cartesian systems of coordinates will be considered: the t, y_r, f- coordinate system of the workpiece and the x, y, z- coordinate sytem of the intersection curve. As shown in Fig.2.1, we define the plane of the intersection curve as the x, y plane. The z axis crosses the f axis of the workpiece at an angle a while the y axis is chosen parallel to the y_r axis. In the following, the intersection curve in the x, y plane will be referred to as the Curve. Positioning a tool in the shape of this Curve above a rotating workpiece at an off-axis distance D and with its plane under an angle a with respect to the t axis, it can be used for production. The shape of the intersection

curve in the x, y plane (the Path) can be derived using Fig.2.2. A point P located on the Curve is given by P = {t_P, y_r,P, f(r)} in the coordinate system of the workpiece or by P = {x_P, y_P, 0} in the coordinate system of the Curve (note that y_r,P = y_P).

Pm t f yr a -4 -2 2 4 -1 1 rm P_D D f t Pm -yr P_D x z -y a Curve g(t, y_r) (a) (b)

Fig.2.1 Functioning principle of Non-circular Ring Surfacing, NRS, showing the position and orientation of the Curve with respect to the surface and the definition of the t, y_r, f system of coordinates of the workpiece and the x, y, z system of coordinates of the Curve: (a) Cross-section of the fabrication configuration in a plane that contains the axis of symmetry of the workpiece, the minimium P_D, the maximum P_m and the t axis. (b) Determination of the Curve for the generation of a certain surface of revolution.

The angle between Curve and workpiece follows from tan( )a = ( )- ( ) ( ) ( ) - = -f r f r f r f t m m P D D D D . (2.1) With r2=t_P2+y_P2 (2.2) and x2=(t_P-D)2+( ( )f r -f( ))D 2 (2.3)

the Path is given by

x r

( )

( )

while z r( ) º 0. Note that f(r) and a are defined as positive for concave surfaces (as seen in the negative f direction) and have negative values for convex surfaces [Figs.2.1 and 2.2].

t P f x x f(r) t a _yr r D z y y r f(D) a P P P

Fig.2.2 Coordinates of point P located on the Curve.

The position of the point P in the x, y plane can also be described by its r_c, f_c coordinates, where r_c is the radial distance of P to the point C = {x_m/2, 0, 0} of the Curve (C in the x, y, z coordinate system) and fc is its angular position in the x, y plane [Fig.2.3]. The coordinates of P then follow from

r_{c( )r = y r( ) +}æ_{x r( )-}xm è öø 2 2 2 , (2.6) f_c m r y r x r x ( ) arctan ( ) ( ) = -é ë ê ê ê ù û ú ú ú 2 , (2.7) with x_m=

(

)

(

)

Thus, the Path, together with the off-axis parameter D and the angle a between Curve and workpiece, determine the shape of the generated surface. Consequently, it is possible to generate with the same Path different surface shapes by changing D and/or a. An example of this will be described in Section 2.4. x /2 x x P r f y C m c c m Path

Fig2.3 Definition of the coordinates r_c, f_c: the Path and its point P in the x, y plane.

For the generation of on-axis surfaces (D = 0) the y_r axis becomes identical with the y axis and the Path is given by

x r

( )

( )

f a a a c m r r f r f r x

( ) arctan sin ( )cos

( ) sin =

(

)

(

)

Fig.2.4# shows three different surfaces together with the therefore required Paths.

2.2 Functioning principle

NRS can be applied with different material removal processes. This section discusses the specific characteristics of the fabrication process that all fabrication techniques based on NRS have in common.

During the fabrication process, a tool that applies an abrasive contact with the workpiece along a two-dimensional form in the shape of the Curve (Path) is positioned above the rotating workpiece such that the Path is located in a plane parallel to the x, y plane at the required off-axis distance D from the f axis. In order to ensure that each point of the tool gets into contact with the whole workpiece surface, the tool is circulating along the Path. In case that, e.g., the Path is an ellipse, the tool is rotating around the axis z_r that is perpendicular to the x, y plane and that contains the point C located on the x axis [Fig.2.5(a)]. The generation of cycloid patterns on the surface is avoided by varying randomly the ratio of the workpiece rotational frequency w1 and the tool circulation frequency w₂ in a small interval. The tool has one translational degree of freedom, such that the Path proceeds parallel to the f axis into the material maintaining its lateral relative position and orientation with respect to the workpiece. The fabrication process ends when the calculated Curve is reached and the required workpiece specifications, the thickness of the optical component, the surface roughness, the amount of sub-surface damage etc., are achieved. NRS is characterized by its three degrees of freedom: the rotation of the workpiece, the rotation of the tool and the translation of the Path [Fig.2.5(a)].

________________________________________

#

In later pictures, where appropriate we will display the Path, surface and surface cross-section in the same order as given in Fig.2.4.

1 3 x -2 -1 1 2 y(x) -4 -2 2 4r 0.2 0.4 f(r) -4 -2 0 2 4 -4 -2 0 2 4 y 0 0.2 f t (a) (b) -5 0 5 -5 0 5 y -35 -25 f t -6 -2 2 6r -30 -20 f(r) 5 15 x -1 -0.5 0.5 1 y(x) -2 -1 1 2r 5 10 15 f(r) 4 8 12 x -0.5 0.5 y(x) -2 0 2 (c) -2 0 2 y 5 10 f t Path surface surface cross-section

Fig2.4 Generation of three different surfaces: (a) concave on-axis ellipsoidal surface with

D = 0, rm = 4, a = 7.26 o

, f(r) = 2 (1 - (1 - r2 / 36)1/2) and y(x) = 3 (-0.04 x (3 x - 12.1))1/2; (b) convex off-axis paraboloidal surface with D = 4, r_m = 6, a = -84.3o, f(r) = -r2and y(x) = 0.1 (20.1 x - x2₎1/2 _{and (c) concave off-axis quartic surface with D = 1, r}

m = 2, a = 86.2 o_{, f(r) =} r4and y(x) = (-(1 + 0.067 x)2 + 0.067 (226 + 225.499 x)1/2).

(a) ( b ) f, w z , w D t y z a x y _r x C m r m 2 r 1 cc a f t r D m z₁ 1 cc₁ z₂ 2 cc₂ a 1 D2

Fig2.5 Cross-section of the fabrication configuration of NRS in the t, f plane: (a) showing its three degrees of freedom: the rotation w₁ of the workpiece, the circulation w₂ of the tool along the Path and the translation of the Path parallel to the f axis until, as shown, the Curve is reached. The cross-section of the Curve in the t, f plane is indicated by c; (b) showing two possible Paths for the generation of the same radial section of a concave surface of revolution f(r) with D2 = - D1.

From Fig.2.1 and Eqs.(2.4) and (2.5) follows that there are two possible fabrication processes for the generation of a certain off-axis part D £ £r r_m of a convex or concave surface of revolution f(r): one with D > 0 (D₁ ) and a second one with D < 0 (D₂ ) (the origin of the x, y, z system of coordinates is located at O ={D, 0, f(D)} (in t, y_r, f coordinates)) [FŠh98a]. In comparison, the one with D > 0 is characterized by a larger angle a (a₁ > a₂), a smaller tool diameter x_m along the x axis and a Path that has a less circular shape. The choice between these two fabrication processes is a trade off between the size of the tool and the required accuracy propagation factor between tool- and workpiece-shape [Subsec.2.5.2] taking into account that a more circular Path is more versatile in the construction of a machine tool [Sec.2.4.2].

The generation of off-axis surfaces using NRS is characterized by the fact that only the required ring element has to be machined. Thus, it is possible to generate the required ring element separately in advance and to machine only the required off-axis part of the surface. Fabrication processes with D < 0 are preferable owing to their larger range of producible shapes. This characteristic will be explained in the following section.

2.3 Producible surfaces

In this section, the producible surface shapes are analysed showing the possibility of producing optical surfaces performing several functions in a single production step [FŠh98a].

2.3.1 Shape

NRS is a shaping method for the generation of rotationally symmetric surfaces. From the way the Curve is determined for a certain surface to be generated [Fig.2.1] follows that it is only possible to generate surface shapes for which continuous intersection curves in the interval D Ê£Êr £ rm exist. The range of producible shapes can be determined by analyzing Eq.(2.5). A producible surface shape satisfies

r2 f r f 2 0 -é - + ëê ùûú ³ ( ) ( ) tan D _D a , (2.13)

which can be re-written as

tana ³ ( )- ( ) -f r f r D D (D £ r £ rm). (2.14)

Therefore, all surfaces of revolution that can be generated are characterized by the fact that, for D £ r £ rm, the cross-section of the surface in the t, f plane, f(t) = g(t, 0), does not cross the x, y plane.

We distingush two different cases. The production of surface shapes with f(D) < f(rm) (e.g. concave surfaces) is feasible if f(t) is located below the x, y plane satisfying

f t( )£cc t( ) (f(D) < f(rm)) (2.15)

(where cc(t) denotes the cross-section of the Curve in the t, f plane [Fig.2.5(a)]). On the other hand, for the production of surface shapes with f(D) > f(r_m) (e.g. convex surfaces), f(t) has to be located above the x, y plane satisfying

f t( )³cc t( ) (f(D) > f(rm)). (2.16)

From the condition stated in Eq.(2.15) for f(D) < f(rm) and the condition stated in Eq.(2.16) for f(D) > f(r_m) we deduce the following. For the generation of concave and convex surfaces, there are always two fabrication processes possible, being characterized by a positive or a negative D, respectively [Sec.2.2]. Conversely, the generation of surface shapes which for D £ £r r_m have a positive derivative (¶f(r)/¶r ³ 0) and a negative second derivative (¶2f(r)/¶r2 £ 0) and the generation of surface shapes which possess a negative derivative (¶f(r)/¶r £ 0) together with a positive second derivative (¶2f(r)/¶r2³ 0) are only possible for off-axis sections of the surfaces (D ¹ 0) using a fabrication process with D < 0. An example of this is shown in Fig.2.6. The generation of a rotationally symmetric on-axis surface with f(r) = r1/2_, ¶f(r)/¶r > 0 and ¶2

f(r)/¶r2Ê< 0 is impossible [Fig.2.6(a)], but an off-axis section of this surface [Fig.2.6(b)] can

be generated. Therefore, fabrication processes with D < 0 enable the production of a larger range of surfaces.

To demonstrate the range of producible shapes, Fig.2.7 shows the generation of a rotationally symmetric surface with f(D) < f(r_m) satisfying Eq.(2.15) that has both a local maximum and a local minimum in the interval D £ £r r_m.

Using NRS, it is possible to generate optical surfaces of revolution that consist of a combination of radial sections with different shapes in a single fabrication process. Such optical surfaces can e.g. be applied to split an incident beam or to focus parts of one incident beam into various foci. An example is given in Fig.2.8 showing the different parts of the Path and the generated surface shapes. It shows the generation of an on-axis surface with ¶f(r)/¶r ³ 0 satisfying Eq.(2.15) for D £ r £ rm. The surface consists of five radial sections of different shapes. In the first radial section, A (0 £ r £ 1), a concave paraboloidal surface is generated, for which an elliptical Path is required. In the next section, B (r = 1), a vertical surface step is generated. The required elliptical

Path y_v(x) is given by the cross-section of the x, y plane with a cylinder, the symmetry axis of

which is the f axis (note, that the Path would loose contact with the surface at r = 1 if the curvature of y(x) would exceed the one of y_v(x)). Subsequently, section C (1 £ r £ 2), an off-axis part of a surface with f r( ) = r, ¶f(r)/¶r > 0 and ¶2f(r)/¶r2Ê<Ê0 is generated, followed by section D (2 £ r £ 3) where a flat surface is created by a Path that is given by a straight line perpendicular to the x axis. Finally, section E, an off-axis part of a concave elliptical surface with r_m = b (b denoting the length of the symmetry axis of the ellipsoid parallel to the r axis) is generated for which an elliptical Path is required.

(a) -10 -5 5 10 t 1 3 5 f t -y f ( b ) -1 -0.5 0.5 1t 0.2 0.6 1 f -1 0 1 t -1 0 1 y 0.8 0.9 f 0.5 1 1.5 x -0.5 0.5 y

Fig.2.6 The generation of a section D £ £r r_m of a rotationally symmetric surface with

f(r)Ê= r1/2_{, ¶f(r)/¶r > 0 and ¶}2_f(r)/¶r2 _{< 0 is only feasible for D ¹ 0. (a) For D = 0, the x, y}

plane and the surface to be generated have only the points {0, 0, 0} and {r_m, 0, f(r_m)} in

common and production using NRS is impossible. (b) Production process using NRS for

f(r) = r1/2_{, with} D = -0.64, r

m = 1, a = 6.95

o _{and y(x) = (1.52 x - 0.93 x}2 _{+ 0.006 x}3 _{+ 2}

(a) -2 0 2 t -2 0 2 y 0.5 1.5 f ( b ) -2 0 2 t -2 0 2 y 0.5 1.5 f -2 2 t 0.6 1.4 1.8 f 1 2 3 x -2 -1 1 2 y

Fig.2.7 The generation of a radial section of a rotationally symmetric surface that has a

maximum and a minimum in the interval D £ £r r_mwith f(r) = 1+(r 2.5)(r 1.6)(1.4 r

-1.5), D = -0.8, r_m = 2.8, a = 21.1o _{and x}

m = 3.85: (a) determination of the Curve by

intersecting the surface to be generated with the x, y plane, (b) required Path together with surface and surface cross-section.

1 2 3 4 r 2 4 6 8 10 12 f 2.5 5 7.5 10 12.5 15 x -3 -2 -1 1 2 3 y A A B B C C D D E _E

Fig.2.8 The generation of an on-axis optical surface that performs several functions in

one fabrication process with a = 75.96o _{showing the five sections of the surface and the}

required Path consisting of five sections of different shape. Section A (0 £ r £ 1) concave

paraboloidal surface with f(r) = r2_{, f(1) = 1, y(x) = 0.5 (4 - 0.243 x)}1/2 _(x)1/2_{; Section B (r}

= 1) vertical surface step with y(x) = (1 - 0.06 x)1/2_{; Section C (1} £ r £ 2) f(r) = r1/2_{, y(x)}

= 0.02 (81 + 209.5 x + 62.1 x2 _{+ 87.1 x}3 _{+ 14.2 x}4₎1/2_{; Section D (2 £ r £ 3) flat surface}

with f(r) = 3,56 and y(x) is given by a straight line perpendicular to the x axis; Section E (3 £ r £ 4) concave elliptical surface with f(r) = r2_{/(1.53 + (2.35 - r}2₎1/2_{), y(x) = 3.5}

(1-0.015(-8.25 + x)2_.

2.3.2 Size

For an existing Curve, the part of the surface that is generated is determined by the fact that during the fabrication process the Path moves parallel to the f axis into the material [Fig.2.5(a)]. Therefore, it is impossible to generate surface sections with an overhang in the interval D £ £r r_m. The generated surface is determined by analyzing the projection of the Path on the t, y_r plane. The radial distance from the point P to the f axis [Figs.2.2 and 2.9] is given by a function r(t) and follows from [see also Eq.(2.2)]

x t

( )

cosa (2.17)

as